Chinese Puzzle Problem

"how many guests are there?" said the official.
"I don't know," said the cook, "but every two shared a dish of rice, every 3 shared a bowl of broth,
and every 4 a dish of meat."

There were 65 dishes in all
How many guests?

Lets start off letting x = number of guests.

We know that there are three different dishes - rice, broth, and meat.  And we know that the total amount of rice dishes, broth bowls, and meat dishes is equal to 65.

Looking at the first statement, "every two shared a dish of rice," we can deduce that every guest has half a dish of rice.  So x/2 = number of rice dishes.

From the next two statements we can deduce that
x/3 = number of broth bowls and
x/4 = number of meat dishes.

So now we can get an equation using the fact that there are 65 dishes in total.
number of rice dishes + number of broth bowls + number of meat dishes = 65
or
x/2 + x/3 + x/4 = 65

From there we can solve for x, which is the number of guests.

6x/12 + 4x/12 + 3x/12 = 65 (Finding a lowest common denominator)
13x/12 = 65 (adding like terms)
x = (65 * 12)/13
x = 60

We can also solve this problem by guessing and checking

Lets guess that there are 120 guests.  That means that there are 60 dishes of rice, 40 bowls of broth, 30 dishes of meat.

60 + 40 + 30 = 130 which does not equal 65.

But we can notice that 130 is double 65.  so our guess was off by a multiplication of 2.
therefore we know that there are 60 guests.

I chose 120 because the numbers work out nicely, but this should work for other numbers with some rounding.



Extensions:  Does it make a difference that this is a Chinese puzzle?  Is culture important in math?

I don't think in this specific problem it makes a difference that is a Chinese puzzle.  The terminology used was irrelevant in my opinion.  It could have been french fries, burgers, and milkshakes.  

But I do believe that culture is important in math.  I remember in highschool we were doing a section on probability and the teacher did an example of a football game.  It wasn't much of a problem to our class as everyone knew what football was and how it was played, but if there was some European students in the class they might not have fully understood the problem.